When we use a neural network to approximate the Q values, is the Q target a single value?
When we use our network to approximate our Q values,is the Q target a single value?

Yes, the target Q value is a single value if you are just updating a single training example. The loss function of a vanilla DQN for a single experience tuple $(s_t,a_t,r_t,s_{t+1})$ is calculated as $$L(\theta) = [r_t + \gamma \,max\,Q_{a_{t+1}}(s_{t+1},a_{t+1};\theta) - Q(s_t,a_t;\theta)]^2$$ where $r_t + \gamma \,max\,Q(s_{t+1},a_{t+1};\theta)$ is the target Q value. However, when using mini-batch gradient descent, you would have to compute multiple target Q values equivalent to the batch size

During backpropagation, when the weights are updated, does it automatically update the Q values, shouldn’t the state be passed in the network again to update it?

During backpropagation of the loss function, the weights $\theta$ are automatically updated. You do not need to pass in the state again. Because in the first place, you would have computed $Q(s_t,a_t;\theta)$ by passing in the state as input to the neural network. That is how backpropagation works for Deep Q networks.
Training for the DQN is as follows:

Collect experience tuples of $(s_t,a_t,r_t,s_{t+1})$ and store them in a replay buffer.
Sample mini-batch of experiences from the replay buffer.
From these sampled batch of experiences, compute $Q(s_{t+1}.a_{t+1};\theta)$ by passing $s_{t+1}$ into the network and take the Q value with the maximum values
Compute $Q(s_t,a_t;\theta)$ by passing $s_t$ into the network.
Compute the Loss for this experience and propagate the loss back to the network, hence updating the weights.

Also, since the weights have changed after backpropagation, the Q values for the same state would also be updated if you pass the same state in to the network again. Check out this paper as it explains how Deep Q Network works.